‘The role of abstract mathematics in solving large or day-to-day problems is usually not perceived by most. It is seen as too abstract. But when you build abstract mathematics, you build more than just a mathematical theory or proof; you construct a new way of thinking and a new way of perceiving, approaching and solving problems.’
These insights come from Dr Sophie Marques, Senior Lecturer in Mathematics at the University of Stellenbosch (SU). She continues: ‘These techniques, approaches and ways of thinking are to me what abstract mathematics bring to the community behind the scenes of sciences. These important components also filter through into the sciences, and the students we teach.’
She makes a crucial point about interdisciplinary connections: ‘One should not underestimate the influence we have on one another, even without realising the subtle interconnections of the disciplines. I believe the power of thinking creatively and in a new way emerges a lot in our work of abstract mathematics. As an example, mathematics teaches us that we can have an elementary look at a problem by taking the most direct approach. If that does not work, you learn to build a tool that requires a specific level and type of specialisation, and you can apply those tools.’
The human side of mathematics
Dr Marques adds that this is ‘parallel to the very narrow application of what my kind of research does, but is also part of what I call the human side of mathematics.’
To further the concept, Dr Marques founded the Wisaarkhu project in 2019. Wisaarkhu magazine, with contributions from students to teachers and academics, was launched in March 2020. The project was created ‘with assistance of the GBB Rubbi Fund to show that mathematics is a human endeavour, despite its inhumanly pristine appearance. It also serves to encourage people to find ways in which mathematics can empower them to reach their goals, as opposed to just seeing it as a subject only for mathematicians.’
In terms of collaborative work, she adds that ‘NITheCS offers wonderful and important opportunities for scientists to collaborate and come together. It offers interdisciplinary events and collaborations. NITheCS is supportive of innovation, open-minded and creative. To me, this is why it is so successful and impactful throughout South Africa. They know how to identify the drivers of research and the innovators, and bring them together to create amazing projects. I am very happy to be an Associate of NITheCS.’
Dr Marques’ was born in Pau, France, and her love for mathematics began in high school. ‘I started tutoring mathematics at the age of 15 and never stopped. My high school teacher, whose husband was an academic, was the first to point out a mathematics career for me. I registered at the Université de Pau et des Pays de l’Adour (UPPA) for a mathematics and computer science degree. Academics was unknown to me then, and a high school mathematics teaching job seemed to be a good idea to pursue my love for mathematics. I therefore registered for a Masters in preparation for the agrégation (French competitive examination for the French educational system) in Bordeaux. For me, this Masters study quickly filled up gaps in algebra, geometry, analysis, and probability statistics, and gave me the taste of research. After obtaining the French agrégation on my first attempt, my love for mathematics led me to pursue a research Masters and then a PhD in mathematics under the supervision of Prof Boas Erez, and Prof Marco Garuti under the Algant programme.’
‘Research only if it has a clear purpose’
She relates how one thing led to the other to open up a new area of research. ‘Before I completed my PhD, I was appointed as a visiting professor assistant at New York University (NYU) where I spent four years. In New York, Dr Jacob Ward, with whom I worked for a few years before he passed away, contacted me with a problem on ramifications. It involved function fields new to me. We wrote a paper involving non-cyclic representation theory and function fields. In this paper, we discussed a few interesting open problems, of which one was related to the notion of a global standard form for the extensions we studied. When I proposed to discover what this notion would be for cubic, I discovered that cubic extensions of function fields were not properly classified. I convinced Dr Jacob Ward to classify these extensions: we managed to carefully organise this small corner of algebra and it permitted us to work with cubics with much more ease and understanding. This work opened an area of research, because not much is done in that direction. It also created collaborations with other researchers and projects with successful outcomes, such as an S2A3/Rector’s award for my Masters student Mpendulo Cele at SU in 2021.’
Dr Marques moved to South Africa in 2017, where she took a postdoctoral position at UCT before being appointed as Senior Lecturer at Stellenbosch University in 2019. Recently, she has made some significant contributions in terms of near-field and near-vector spaces. ‘Significantly, we found a way to define generalised means for complex numbers.’
This mother of two children insists that ‘I do not believe either in a hierarchy of discipline or skill or hierarchy in general. I do not think my work is more important. But I do think it is important. Mathematics shapes the world deeper than just in terms of technology as we might perceive it. It also shapes the political world, the emotional world, the workforce, and relationships.’
The impact of research
She explains: ‘I am convinced that how we do research in mathematics and how the research is transmitted has an important impact on the way society operates at any level – from individual to community to global. I try my best to undertake research only if it has a clear purpose and impact. I consider many different parameters of thinking, at least in my own eyes, and whether I can logically and scholarly support it by an argument. I try to transmit these values to my students. I believe that only in doing so can we solve the main global problems. If we lose ourselves in competition and the rat race, the solutions found could be harmful in the long term due to a lack of careful consideration.’
Dr Marques advises that young scientists should ‘stay open-minded. True growth is slow and takes time. Be patient with yourself and always be willing to learn. Your knowledge gets limited when you are not willing to question it constantly. Be honest with yourself, and understand that what you don’t understand is also a step further! You are uniquely talented, and this talent cannot be reached by trying to compete with or being someone else. Don’t forget your human side and the human side of your discipline.’